A356253 a(n) is the largest coefficient of P(x) := Product_{k} (x + p_k), where (p_k) are the primes dividing n listed with multiplicity.

1, 2, 3, 4, 5, 6, 7, 12, 9, 10, 11, 16, 13, 14, 15, 32, 17, 21, 19, 24, 21, 22, 23, 44, 25, 26, 27, 32, 29, 31, 31, 80, 33, 34, 35, 60, 37, 38, 39, 68, 41, 42, 43, 48, 45, 46, 47, 112, 49, 50, 51, 56, 53, 81, 55, 92, 57, 58, 59, 92, 61, 62, 63, 240, 65, 66, 67, 72

Offset: 1

Author

Thomas Scheuerle, Jul 31 2022

Keywords

• nonn

Comments

a(n) is the greatest number we may obtain by applying elementary symmetric functions onto the prime factors of n with multiplicity.

The record values of a(n)/n appear at powers of two.

If a(n) is greater than n, then it equals in most cases A003415(n), the first exception where a(n) > A003415(n) > n is at n = 64.

Conjectured: a(A002110(n)) = A024451(n), for n > 2.

Conjecture equality breaks down after n = 175, as a(A002110(176)) > A024451(176). - Antti Karttunen, Feb 08 2024

Links

• Antti Karttunen, Table of n, a(n) for n = 1..30030

Crossrefs

Cf. A002110, A003415, A024451, A070918, A083348, A109388, A260613, A369657 (difference between this sequence and A003415).

Cf. A065048 (same concept but uses numbers 1..n instead of prime factors of n).

Programs

PARI

a(n) = vecmax(Vec(vecprod([(x+f[1])^f[2] | f<-factor(n)~]))) \\ Edited by M. F. Hasler, Feb 14 2024

Formula

a(n) = n iff n is not in A083348, otherwise a(n) > n.

a(2^n) = A109388(n) = binomial( n, floor(n/3) )*2^(n-floor(n/3)).

a(p^n) = binomial( n, floor(n/(p+1)) )*p^(n-floor(n/(p+1))), if p is prime.

a(p*n)/a(n) >= n and <= n+1 if p is prime.

a(p*q)/a(q) = p if p and q are prime. This is also true if q is a prime greater than 7 and p is a product of two primes greater than 4.

a(A002110(n)) >= A024451(n), for n > 2. The maximum of row n in A260613 a variant of A070918.